Modeling the Observations of GRB 180720B: from Radio to Sub-TeV Gamma-Rays

The data files used for this analysis were obtained from the data website.⁷ They contain information 600 s before up to 1000 s from the trigger time (T₀) (2018-07-20 14:21:39.65 UTC; Bissaldi & Racusin 2018). Fermi-LAT data were analyzed in the 0.1–100 GeV energy range and the time interval of T₀ + 10 s up to T₀ + 630 s with the Fermi Science tools⁸ ScienceTools v10r00p05. For this analysis we adopt the P8R2_TRANSIENT020_V6 response, following the unbinned likelihood analysis presented by the Fermi-LAT team.⁹ Using the gtselect tool, we select, with an eventclass 16, a region of interest (ROI) around the position of this burst within a radius of 10°. We apply a cut on the zenith angle above 100°. Then we select the appropriate time intervals (GTIs) using the gtmktime tool on the data selected before considering the ROI cut. In order to define the model needed to describe the source and the diffuse components, we use modeleditor. We define a point source at the position of this burst, assuming a power-law spectrum, and we define a galactic diffuse component using GALPROP gll_iem_v06 as well as the extragalactic background iso_P8R3_SOURCE_V2.¹⁰ We use gtdiffrsp to take into account all of these components. Following the likelihood procedure, we produce a lifetime cube with the tool gtltcube, using a step δθ = 0.025, a bin size of 0.5 and a maximum zenith angle of 100°. The exposure map was created using gtexpmap, considering a region of 30° around the GRB position and defining 100 spatial bins in longitude/latitude and 50 energy bins. Finally, we perform the likelihood analysis with gtlike, obtaining a photon flux of (5.2 ± 0.4) × 10⁻⁵ photons cm⁻² s⁻¹ and a test statistic of 883.267.

We find photons with a probability greater than 90% of being associated with GRB 180720B using the gtsrcprob. In this case, we use gtbin in order to obtain the light curve, considering eight logarithmically uniform temporal bins. The photon flux is generated using the counts and the exposure in each bin. The exposure is obtained with gtexposure. To derive the energy flux we compute the spectra integrated over each interval assuming logarithmic binning for the energy between 100 MeV and 100 GeV. Then we obtain the Detector Response Matrix with the gtrspgen tool assuming a point=like source, a maximum cutoff angle of 60°, and a bin size of 0.05 into 30 logarithmically uniform bins between 100 and 100 GeV. Finally, we derive the background spectra using gtbkg and subtract it to the source using XSPEC (v12.10.1; Arnaud 1996) in order to obtain the energy flux with a 90% confidence error.

The left panel of Figure 1 displays the Fermi-LAT energy flux (blue) and photon-flux (red) light curves (upper panel) and all the photons with energies >100 MeV associated with GRB 180720B (lower panel). The filled circles in the bottom panel correspond to the individual photons and their energies with a >0.9 probability of being associated with GRB 180720B and the open circles indicate the LAT gamma Transient class photons. The dotted and dashed lines on the photon-flux light curve correspond to the best-fit curves using a power-law (PL) function and a broken power-law (BPL) function. The best-fit values are α = 1.81 ± 0.16 for the PL function and α₁ = 1.49 ± 0.12 *α₂ = 3.09 ± 0.64 for the BPL function. These values are consistent with the ones from the photon-flux light curve reported in Table 5 of Ajello et al. (2019). The right panel of Figure 1 shows the Fermi-LAT spectrum.

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We modeled the energy flux light curve and spectrum using the closure relation ${F}_{\nu ,f}^{\mathrm{syn}}\propto {t}^{-{\alpha }_{\mathrm{LAT}}}{\epsilon }_{\gamma }^{-{\beta }_{\mathrm{LAT}}}$. The best-fit values of the temporal and spectral PL indexes were α_LAT = 1.45 ± 0.53 (χ² = 1.11) and β_LAT = 1.17 ± 0.15 (χ² = 1.09), respectively. These PL indexes are compatible with the third PL segment of the synchrotron forward-shock model $\left(\propto {t}^{-\tfrac{3p-2}{4}}{\epsilon }_{\gamma }^{-\tfrac{p}{2}}\right)$ for p ≈ 2.6 ± 0.2. It is worth emphasizing that this PL segment is equal for the wind and homogeneous afterglow model.

Some relevant characteristics can be observed in the lower panel of Figure 1: (i) the first HE photon (101 MeV) was detected 19.4 s after the trigger time, (ii) this burst exhibited 130 photons with energy greater than 100 MeV and 8 photons with energies greater than 1 GeV, (iii) the highest-energy photon¹¹ exhibited in the LAT observations (4.9 GeV) was detected 142.43 s after the trigger time, and (iv) the photon density increased dramatically for a time longer than ≳50 s.

The Fermi-GBM data were obtained using the public database at the GBM website.¹² The event data files were obtained using the Fermi-GBM Burst Catalog¹³ and the GBM trigger time for GRB 180720B at 14:21:39.65 UT (Roberts & Meegan 2018). Flux values were derived using the spectral analysis package Rmfit version 432.¹⁴ In order to analyze the signal we used the time-tagged event files of the two triggered NaI detectors n₇ and n₁₁ and the BGO detector b₁. Two different models were used to fit the spectrum in the energy range of 10–1000 keV over different time intervals. The Band and the Comptonized models were used to fit the spectrum during the time interval [0.000, 60.416 s]. Each time bin was chosen by adopting the minimum resolution required to preserve the shape of the time resolution.

The upper left panel in Figure 2 displays the GBM light curve in the 10–1000 keV energy range. This light curve shows a bright, fast-rise exponential-decay (FRED)-like peak with a maximum flux of $2.74\times {10}^{-5}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ at 15 s, followed by two significant peaks with fluxes of 1.64 × 10⁻⁵ and $5.5\times {10}^{-6}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ at 26 s and 50 s, respectively. The fluence over the prompt emission was (2.985 ± 0.001) × 10⁻⁴ erg cm⁻², which corresponds to an equivalent isotropic energy of 3 × 10⁵³ erg for a measured redshift of z = 0.654 (Vreeswijk et al. 2018). This light curve exhibits a high-variability δt/t ≪ 1,¹⁵ which favors the prompt phase scenario. Theoretically, this timescale is interpreted as the time difference of two photons emitted at two different radii (Sari & Piran 1997).

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The Swift BAT triggered on GRB 180720B on 2018 July 20 at 14:21:44 UT. This instrument located this burst at the coordinates: R.A. = 00^h02^m07^s and decl. = −02^d56'00'' (J2000), with an uncertainty of 3'. The XRT instrument started observing this burst 86.5 s after the trigger, and monitored the afterglow for the following 33.5 days, giving a total net exposure of 13 ks in Windowed Timing (WT) mode and 2.8 × 10³ ks in Photon Counting (PC) mode. The Swift data used in this analysis are publicly available in the website database.¹⁶ In the WT mode, the reported value of the photon spectral index was ${{\rm{\Gamma }}}_{{\rm{X}}}\,={\beta }_{{\rm{X}}}+1=1.761\pm 0.01$ for a galactic (intrinsic) absorption of ${N}_{{\rm{H}}}=3.92(17.7\pm 0.7)\times {10}^{20}\,{\mathrm{cm}}^{-2}$. In the PC mode, the reported value of the photon spectral index was ${{\rm{\Gamma }}}_{{\rm{X}}}={\beta }_{{\rm{X}}}+1=1.83\pm 0.06$ for ${N}_{{\rm{H}}}=3.92(24.0\pm 0.4)\,\times {10}^{20}\,{\mathrm{cm}}^{-2}$.

The upper right panel in Figure 2 shows the Swift X-ray light curve obtained with the XRT (WC and PC modes) instrument at 1 keV. The flux density of the XRT data was extrapolated from 10 to 1 keV using the conversion factor introduced in Evans et al. (2010). The blue curves correspond to the best-fit PL functions obtained using the chi-square minimization algorithm installed in ROOT (Brun & Rademakers 1997). In accordance with the observational X-ray data, three PL segments (${t}^{-{\alpha }_{{\rm{X}}}}$) with an X-ray flare were identified in this light curve. We evaluated the X-ray light curve at four time intervals, designated as epochs I, II, III, and IV: 70 ≲ t ≲ 200 s (I), 200 ≲ t ≲ 2.5 × 10³ s (II), 2500 ≲ t ≲ 2.6 × 10⁵ s (III) and t ≥ 2.6 × 10⁵ (IV). The time intervals were chosen in accordance with the variations of each slope. The temporal PL indexes are ${\alpha }_{{\rm{X}},\mathrm{rise}}=-2.05\,\pm 0.27({\chi }^{2}/\mathrm{ndf}=1.12)$ and ${\alpha }_{{\rm{X}},\mathrm{decay}}=2.74\pm 0.08\,(1.27)$ during epoch "I" and α_X = 0.79 ± 0.06(1.31), 1.26 ± 0.06(1.29) and 1.75 ± 0.09(1.21) for epochs "II," "III," and "IV," respectively. The best-fit values of each epoch are reported in Table 1.

GRB 180720B began to be detected in the optical and near-infrared bands on 2018 July 20 at 14:22:57 UT, 73 s after the trigger time (Sasada et al. 2018). Using the HOWPol and HONIR instruments attached to the 1.5 m Kanata telescope, these authors reported a bright optical R-band counterpart of m_R = 9.4 mag. Vreeswijk et al. (2018) observed the optical counterpart of this burst using the VLT/X-shooter spectrograph. They detected a bright continuum with some absorption lines (Fe ii, Mg ii, Mg i, and Ca ii) associated with a redshift of z = 0.654. Additional photometry in different optical bands is reported in Martone et al. (2018), Reva et al. (2018), Itoh et al. (2018), Crouzet & Malesani (2018), Horiuchi et al. (2018), Watson et al. (2018), Schmalz et al. (2018), and Lipunov et al. (2018).

The lower panel in Figure 2 shows the optical light curve of GRB 180720B in the R-band. The solid line represents the best-fit PL function. Optical data taken from the GCN circulars reported in this subsection were detected by different telescopes. The optical observations with their uncertainties were obtained using the standard conversion for AB magnitudes shown in Fukugita et al. (1996). The optical data were corrected by the galactic extinction using the relation derived in Becerra et al. (2019b). The values of β_O = 0.80 ± 0.04 for optical filters and a reddening of ${E}_{B-V}=0.037$ mag (Bolmer & Shady 2019) were used. The best-fit value of the temporal decay is 1.22 ± 0.02 (${\chi }^{2}/\mathrm{ndf}=1.05;$ see Table 2).

Sfaradi et al. (2018) observed the position of this burst with AMI-LA at 15.5 GHz for 3.9 hr. The observations began 2 days after the BAT trigger, providing an integrated flux of ∼1 mJy. Chandra et al. (2018) detected GRB 180720B with GMRT at the 1.4 GHz band, reporting a flux of ∼390 ± 59 μJy.

During the Cerenkov Telescope Array (CTA) Science Symposium 2019, the HESS collaboration reported the discovery of late-time VHE emission from GRB 180720B. The VHE emission with ∼5σ was in the energy range from 100 to 400 GeV. The observations began ∼10 hr after the burst trigger.¹⁷

During this epoch, the LAT and the optical light curves are modeled with PL functions and the X-ray flare is modeled with two PLs. Considering that during this epoch the X-ray flare, the LAT and the optical light curves have different origins, we first analyze the LAT and the optical light curves and then we examine the X-ray flare.

3.1.1. Analysis of LAT and Optical Light Curves

3.1.2. Analysis of the X-Ray Flare

During this epoch, the spectral analysis indicates that the optical and X-ray observations are described with a PL with an index ${\beta }_{{\rm{X}},\mathrm{III}}=0.70\pm 0.05$. The temporal analysis shows that the indexes of optical (${\alpha }_{{\rm{O}}}=1.22\pm 0.02$) and X-ray (${\alpha }_{{\rm{X}},\mathrm{III}}=1.26\pm 0.06$) observations are consistent. Therefore, the optical and X-ray fluxes evolve in the second PL segment of synchrotron emission in the homogeneous medium for predicted values of p ≈ 2.62 ± 0.02 and p ≈ 2.68 ± 0.08, respectively. In the context of the X-ray light curve, this phase is known as the normal decay (e.g., see Zhang et al. 2006).

The X-ray light curve during this time interval decays with ${\alpha }_{{\rm{X}},\mathrm{IV}}=1.70\pm 0.19$, which is consistent with the LAT light curve reported in epoch II. Taking into account epoch III, the temporal PL index varied as Δα ≈ 0.45 (from ${\alpha }_{{\rm{X}},\mathrm{II}}\,=1.26\pm 0.06$ to 1.70 ± 0.19), which is consistent with the evolution from the second to the third PL segments of synchrotron emission in the homogeneous medium for p ≈ 2.60 ± 0.2. Therefore, the break observed in the X-ray light curve during the transition from epochs III to IV can be explained as the transition of the synchrotron energy break below the X-ray observations at 1 keV.

In order to describe the LAT, X-ray, and optical light curves correctly during this time interval, epochs I and III are taken into account. Temporal and spectral analysis of the X-ray light curve shows that during epoch II the PL indexes are ${\alpha }_{{\rm{X}},\mathrm{II}}\,=0.79\pm 0.08$ and ${\beta }_{{\rm{X}},\mathrm{II}}=0.68\pm 0.06$, respectively. The spectral indexes associated with the X-ray observations during epochs II and III are very similar (${\beta }_{{\rm{X}},\mathrm{II}}\approx {\beta }_{{\rm{X}},\mathrm{III}}$), and the spectral and temporal PL indexes of LAT and optical observations during epochs I and II are unchanged. Taking into account that ${\beta }_{{\rm{X}},\mathrm{II}}\approx {\beta }_{{\rm{X}},\mathrm{III}}$, that there are no breaks in the LAT and optical light curves, and that the value of the temporal decay index is followed by the normal decay phase in the X-ray light curve, epoch II is consistent with the shallow "plateau" phase (e.g., see Vedrenne & Atteia 2009). It is worth highlighting that during this transition there was no variation of the spectral index. A priori, we could think that X-ray observations during this epoch could be associated with the second PL segment $\left(\propto {t}^{-\tfrac{(3p-3)}{4}}{\epsilon }_{\gamma }^{-\tfrac{p-1}{2}}\right)$ of synchrotron emission. In this case the spectral index of the electron population, taking into account the temporal and spectral analysis, would be p = 2.05 ± 0.11 and p = 0.53 ± 0.11, respectively, which is different from the LAT and optical observations derived in the previous subsection. Hence, this hypothesis is rejected and we postulate the "plateau" phase.

The temporal and spectral theoretical indexes obtained by the evolution of the standard synchrotron model in the homogeneous medium are reported in Table 2. Theoretical and observational spectral and temporal indexes are consistent for p ≈ 2.6 ± 0.2.

During the afterglow, the self-absorption energy break lies in the radio bands and falls into two groups depending on the regime (fast- or slow-cooling) of the electron energy distribution. For $\min \{{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}},{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}\}\lt {\epsilon }_{{\rm{c}},{\rm{f}}}^{\mathrm{syn}}$, the observed synchrotron radiation flux can be in the weak (${\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}\leqslant {\epsilon }_{\gamma }\leqslant {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}$) or strong (${\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}\leqslant {\epsilon }_{\gamma }\leqslant {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}$) absorption regime (e.g., see, Gao et al. 2013).

Weak-absorption regime. In this case the synchrotron spectrum in the radio bands can be written as

$$\begin{eqnarray}{F}_{\nu ,f}^{\mathrm{syn}}={F}_{\nu ,\max }^{\mathrm{syn}}\left\{\begin{array}{ll}{\left(\tfrac{{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}}{{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}}\right)}^{\tfrac{1}{3}}{\left(\tfrac{{\epsilon }_{\gamma }}{{\epsilon }_{{\rm{a}}}^{\mathrm{syn}}}\right)}^{2}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}},\\ {\left(\tfrac{{\epsilon }_{\gamma }}{{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}}\right)}^{\tfrac{1}{3}} & {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}},\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where the self-absorption energy break is

$$\begin{eqnarray}&&{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}\propto {(1+z)}^{-\tfrac{8}{5}}{\epsilon }_{{\rm{e}}}^{-1}\,{\epsilon }_{{\rm{B}},{\rm{f}}}^{\tfrac{1}{5}}{{\rm{\Gamma }}}^{\tfrac{8}{5}}\,{n}^{\tfrac{4}{5}}\,{t}^{\tfrac{3}{5}}.\end{eqnarray} \tag{ 2 }$$

From Equations (1) and (2), the radio light curve becomes ${F}_{\nu ,f}^{\mathrm{syn}}\propto :$ ${t}^{\tfrac{1}{2}}{\epsilon }_{\gamma }^{2}$ for ${\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}$ and ${t}^{\tfrac{1}{2}}{\epsilon }_{\gamma }^{\tfrac{1}{3}}$ for ${\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}$.¹⁸

Strong-absorption regime. In this case, the synchrotron spectrum in the radio bands is

$$\begin{eqnarray}{F}_{\nu ,f}^{\mathrm{syn}}={F}_{\nu ,\max }^{\mathrm{syn}}\left\{\begin{array}{ll}{\left(\tfrac{{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}}{{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}}\right)}^{\tfrac{p+4}{2}}{\left(\tfrac{{\epsilon }_{\gamma }}{{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}}\right)}^{2}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}},\\ {\left(\tfrac{{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}}{{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}}\right)}^{-\tfrac{p-1}{2}}{\left(\tfrac{{\epsilon }_{\gamma }}{{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}}\right)}^{\displaystyle \frac{5}{2}} & {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}},\end{array}\right.\end{eqnarray} \tag{ 3 }$$

where the self-absorption energy break in this case is

$$\begin{eqnarray}&&{\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}\propto {(1+z)}^{-\tfrac{p+6}{p+4}}{\epsilon }_{{\rm{e}}}^{-\tfrac{2(p-1)}{p+4}}\,{\epsilon }_{{\rm{B}},{\rm{f}}}^{\tfrac{p+2}{2(p+4)}}{{\rm{\Gamma }}}^{\tfrac{4(p+2)}{p+4}}\,{n}^{\tfrac{p+6}{2(p+4)}}\,{t}^{\tfrac{2}{p+4}}.\end{eqnarray} \tag{ 4 }$$

From Equations (3) and 4, the radio light curve becomes ${F}_{\nu ,f}^{\mathrm{syn}}\propto :$ ${t}^{\tfrac{1}{2}}{\epsilon }_{\gamma }^{2}$ for ${\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}$ and ${t}^{\tfrac{5}{4}}{\epsilon }_{\gamma }^{\tfrac{5}{2}}$ for ${\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}$.

Taking into account the best-fit values reported in Table 3, the synchrotron energy breaks are ${\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}=25.43\,\mathrm{GHz}$, ${\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}\,=13.6\,\mathrm{GHz}$ (weak absorption) and ${\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}=2.4\times {10}^{3}\,\mathrm{GHz}$ (strong absorption) at 1 day, and ${\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}=0.8\,\mathrm{GHz}$, ${\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}=13.6\,\mathrm{GHz}$ (weak absorption) and ${\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}=4.5\times {10}^{2}\,\mathrm{GHz}$ (strong absorption) at 10 days. Clearly, the synchrotron spectrum lies in the regime ${\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{syn}}\leqslant {\epsilon }_{\gamma }\leqslant {\epsilon }_{{\rm{a}},{\rm{f}}}^{\mathrm{syn}}$ and radio observations are in the second PL segment.

Using the best-fit parameters obtained with our MCMC code, we describe the radio data points as shown in the right panel of Figure 4. In order to describe the radio data at 15.5 and 1.4 GHz with a PL, we multiply the radio data point at 1.4 GHz by 25 and we also normalize the synchrotron flux at the same radio band.

Using the best-fit value of the homogeneous density found with our MCMC code. in Figure 8 we plot the evolution of the maximum photon energy radiated by synchrotron emission from the forward-shock region (red dashed line) and all individual photons with probabilities >90% of being associated with GRB 180720B and energies above ≳100 MeV. Photons with energies above the synchrotron limit are in gray and ones with energy below this limit are in black. The sensitivities of CTA and HESS-CT5 observatories at 75 and 80 GeV are shown as yellow and blue dashed lines, respectively (Piron 2016). The emission reported by the HESS Collaboration during the CTA symposium¹⁹ is shown in green. Figure 8 shows that all photons cannot be interpreted in the standard synchrotron forward-shock model. Although this burst is a good candidate for accelerating particles up to very high energies and then producing TeV neutrinos, no neutrinos were spatially or temporally associated with this event. This negative result could be explained in terms of the low amount of baryon load in the outflow. In this case the production of VHE photons favors leptonic over hadronic models. Therefore, we propose that LAT photons above the synchrotron limit would be interpreted in the SSC framework. Note that LAT photons below the synchrotron limit (the red dashed line) could be explained in the standard synchrotron forward-shock model and beyond this limit the SSC model describes the LAT photons. For instance, a superposition of synchrotron and SSC emission originated in the forward-shock region could be invoked to interpret the LAT photons (e.g., see Beniamini et al. 2015).

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The Fermi-LAT photon-flux light curve of GRB 180720B presents characteristics similar to other LAT-detected GRBs, such as GRB 080916C (Abdo et al. 2009b), GRB 090510 (Ackermann et al. 2010), GRB 090902B (Abdo et al. 2009a), GRB 090926A (Ackermann et al. 2011) GRB 110721A (Ackermann et al. 2013a), GRB 110731A (Ackermann et al. 2013a), GRB 130427A (Ackermann et al. 2014), GRB 160625B (Fraija et al. 2017c), and GRB 190114C (Fraija et al. 2019a), as shown in Figure 9. All of these GRBs exhibited VHE photons and emission lasting more than the prompt phase. This figure shows that during the prompt phase, the high-energy emission from GRB 180720B is weaker and during the afterglow it is stronger.

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The dynamics of the synchrotron forward-shock emission in a homogeneous medium have been widely explored (e.g., see, Sari et al. 1998). Synchrotron photons radiated in the forward shocks can be upscattered by the same electron population. The inverse Compton scattering model has been described in Panaitescu & Mészáros (2000) and Kumar & Piran (2000). Given the energy breaks, the maximum flux, the spectra, and the light curves of the synchrotron radiation, the SSC light curves for the fast-cooling and slow-cooling regimes become

$$\begin{eqnarray}{F}_{\nu ,f}^{\mathrm{ssc}}\propto \left\{\begin{array}{ll}{t}^{\tfrac{1}{3}}\,{\epsilon }_{\gamma }^{\tfrac{1}{3}}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}},{\rm{f}}}^{\mathrm{ssc}},\\ {t}^{\tfrac{1}{8}}\,{\epsilon }_{\gamma }^{-\tfrac{1}{2}}, & {\epsilon }_{{\rm{c}},{\rm{f}}}^{\mathrm{ssc}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{ssc}},\\ {t}^{-\tfrac{9p-10}{8}}\,{\epsilon }_{\gamma }^{-\tfrac{p}{2}}, & {\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{ssc}}\lt {\epsilon }_{\gamma },\end{array}\right.\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}\begin{array}{r}{F}_{\nu ,f}^{{\rm{s}}{\rm{s}}{\rm{c}}}\propto \left\{\begin{array}{ll}t\,{\epsilon }_{\gamma }^{{\textstyle \tfrac{1}{3}}}, & {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{m}},{\rm{f}}}^{{\rm{s}}{\rm{s}}{\rm{c}}},\\ {t}^{-{\textstyle \tfrac{9p-11}{8}}}{\epsilon }_{\gamma }^{-{\textstyle \tfrac{p-1}{2}}}, & {\epsilon }_{{\rm{m}},{\rm{f}}}^{{\rm{s}}{\rm{s}}{\rm{c}}}\lt {\epsilon }_{\gamma }\lt {\epsilon }_{{\rm{c}},{\rm{f}}}^{{\rm{s}}{\rm{s}}{\rm{c}}},\\ {t}^{-{\textstyle \tfrac{9p-10}{8}}}\,{\epsilon }_{\gamma }^{-{\textstyle \tfrac{p}{2}}}, & {\epsilon }_{{\rm{c}},{\rm{f}}}^{{\rm{s}}{\rm{s}}{\rm{c}}}\lt {\epsilon }_{\gamma },\end{array}\right.\end{array}\end{eqnarray} \tag{ 6 }$$

respectively, where the SSC energy breaks are

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{{\rm{m}},{\rm{f}}}^{\mathrm{ssc}} & \propto & {(1+z)}^{\tfrac{5}{4}}\,{\epsilon }_{{\rm{e}}}^{4}\,{\epsilon }_{{\rm{B}},{\rm{f}}}^{\tfrac{1}{2}}\,{n}^{-\tfrac{1}{4}}\,{E}^{\tfrac{3}{4}}\,{t}^{-\tfrac{9}{4}}\\ {\epsilon }_{{\rm{c}},{\rm{f}}}^{\mathrm{ssc}} & \propto & {(1+z)}^{-\tfrac{3}{4}}{(1+Y)}^{-4}\,{\epsilon }_{{\rm{B}},{\rm{f}}}^{-\tfrac{7}{2}}\,{n}^{-\tfrac{9}{4}}\,{E}^{-\tfrac{5}{4}}\,{t}^{-\tfrac{1}{4}},\end{array}\end{eqnarray} \tag{ 7 }$$

with Y being the Compton parameter. In the Klein–Nishina regime the break energy is given by

$$\begin{eqnarray}&&{E}_{{\rm{c}}}^{\mathrm{KN}}\propto {(1+z)}^{-1}\,{(1+Y)}^{-1}\,{\epsilon }_{{\rm{B}},{\rm{f}}}^{-1}\,{n}^{-\tfrac{2}{3}}\,{{\rm{\Gamma }}}^{\displaystyle \frac{2}{3}}\,{E}^{-\tfrac{1}{3}}\,{t}^{-\tfrac{1}{4}}.\end{eqnarray} \tag{ 8 }$$

In this case the maximum flux emitted in this process is ${F}_{\nu ,\max }^{\mathrm{ssc}}\propto {(1+z)}^{\tfrac{3}{4}}\,{\epsilon }_{{\rm{B}},{\rm{f}}}^{\tfrac{1}{2}}\,{n}^{\tfrac{5}{4}}\,{d}^{-2}\,{E}^{\tfrac{5}{4}}\,{t}^{\tfrac{1}{4}},$ where d is the luminosity distance.

Given the best-fit parameters reported in Table 3, the SSC light curve at 100 GeV is plotted in Figure 4 (right). The break energy in the KN regime is 851.7 and 478.9 GeV at 1 and 10 hr, respectively, which is above the flux at 100 GeV. The characteristic and cutoff break energies are 4.4 × 10³ and 24.7 eV, and 50.1 and 43.2 GeV at 1 and 10 hr, respectively, which indicates that at 100 GeV, the SSC emission is evolving in the third PL segment of the slow-cooling regime. The right panel in Figure 4 shows the SSC flux computed at 100 GeV with the parameters derived in our model using the multiwavelength observations of GRB 180720B. Our model explains the VHE emission announced by the HESS team during the high-energy emission at the CTA symposium.²⁰ Note that the SSC flux equations are degenerate in the parameter values such that for a distinct set of parameter values similar results could be obtained, as shown in Wang et al. (2019).

The best-fit parameters of the magnetic fields found in the forward- and reverse-shock regions are different. The parameter associated with the magnetic field in the reverse shock lies in the range of the expected values for the reverse shock to be formed and leads to an estimate of the magnetization parameter of σ ≃ 0.4. In the opposite situation (e.g., σ ≫ 1), particle acceleration would hardly be efficient and the X-ray flare from the reverse shock would have been suppressed (Fan et al. 2004). Considering the microphysical parameter associated with the reverse-shock region, we found that the strength of the magnetic field in this region is stronger than the magnetic field in the forward-shock region (≃45 times). This suggests that the jet composition of GRB 180720B could be Poynting-dominated. Zhang & Kobayashi (2005) described the emission generated in the reverse shock from an outflow with an arbitrary value of the magnetization parameter. They found that the Poynting energy is transferred to the medium only until the reverse shock has disappeared. Given the timescale of the reverse shock associated with the X-ray flare, the shallow decay segment observed in the X-ray light curve of GRB 180720B could be interpreted as the late transfer of the Poynting energy to the homogeneous medium. These results agree with some authors who claim that Poynting flux-dominated models with a moderate degree of magnetization can explain the LAT observations in powerful GRBs (Zhang & Yan 2011; Uhm & Zhang 2014), and in particular the properties exhibited in the light curve of GRB 180720B.

Using the synchrotron reverse-shock model (Kobayashi 2000; Kobayashi & Zhang 2003) and the best-fit values (see Table 3), the self-absorption, the characteristic and cutoff energy breaks of 3.4 × 10⁻⁷ eV, 0.4 eV and 7.3 × 10⁻⁴ eV, respectively, indicate that the synchrotron radiation evolves in the fast-cooling regime. Given that the self-absorption energy break is smaller than the cutoff and characteristic breaks, the synchrotron emission originated from the reverse-shock region lies in the weak self-absorption regime, hence a thermal component in this region cannot be expected (Kobayashi & Zhang 2003).

The spectral and temporal analysis of the forward and reverse shocks at the beginning of the afterglow phase, together with the best-fit value of the circumburst density, lead to an estimate of the initial bulk Lorentz factor, the critical Lorentz factor, and the shock crossing time Γ ≃ 200, Γ_c ≃ 330 and t_d ≃ 100 s, respectively. These values are consistent with the evolution of the reverse shock in the thin shell case and the duration of the X-ray flare.